Generalized graph splines and the Universal Difference Property

We study the generalized graph splines introduced by Gilbert, Tymoczko, and Viel and focus on an attribute known as the Universal Difference Property (UDP). We prove that paths, trees, and cycles satisfy UDP. We explore UDP on graphs pasted at a single vertex and use Prüfer domains to illustrate that not every edge labeled graph satisfies UDP. We show that UDP must hold for any edge labeled graph over a ring $R$ if and only if $R$ is a Prüfer domain. Lastly, we prove that UDP is preserved by isomorphisms of edge labeled graphs.